## Introduction

The searches for the exotic spin-dependent interactions between fermions1 have recently attracted attention2. To date, 15 possible interactions have been derived by considering the spins and the relative velocity of two interacting fermions1,3. In recent theoretical literature, mediators of such interactions have been generically defined as WISPs (weakly interacting sub-eV particles)4 including the most plausible mediators, the axion and the hidden photon. The axion is a hypothetical spin-0 pseudoscalar boson that can explain the puzzling strong charge-parity (CP) problem in the quantum chromodynamics (QCD)5,6,7 and is a theoretically well-motivated candidate for dark matter in the Universe8,9. The hidden photon10,11,12 is a hypothetical spin-1 boson arising from the breaking of a U(1) gauge symmetry, which can be a dark matter candidate as well, indirectly interacting with ordinary particles13,14. Detection of WISPs is challenging because they are only very weakly coupled to fermions to mediate interactions between the fermions. Recent WISPs searches, such as the Axion Dark Matter eXperiment (ADMX)15, are reviewed in ref. 16.

Searches for exotic interactions have been typically conducted by closely positioning two objects and using a sensitive detector such as a superconducting quantum interference device (SQUID)17, electron-spin polarized torsion-pendulum18, or precessing nuclear spins in nuclear magnetic resonance (NMR)19. Recently, we have expanded this area of research using a highly sensitive non-cryogenic atomic magnetometer based on an optically pumped alkali-metal atomic vapor20,21. The atomic magnetometer operates in the spin-exchange relaxation-free (SERF) regime in which the effects of spin-exchange collisions on spin relaxation are tuned off, dramatically improving the sensitivity to the femto-Tesla level22,23. Our approach probes exotic interactions between optically polarized electrons in an atomic vapor and unpolarized or polarized particles of a solid-state mass in the vicinity of the atomic vapor.

Since the first parity-odd interaction has been discovered in the weak interaction sector24,25, various searches for parity-odd interactions have been performed, for example, atomic parity non-conservation (PNC) experiments26 and electric dipole moment (EDM) experiments27,28. The searches could be pivotal for developing theories beyond the Standard Model of particle physics. In the present study we investigate an exotic parity-odd spin- and velocity-dependent interaction of electrons with axial-vector electron coupling and vector nucleon coupling $$\left( {g_{\mathrm{A}}^{\mathrm{e}}g_{\mathrm{V}}^{\mathrm{N}}} \right)$$3, where A, V, e, and N stand for the axial-vector coupling, the vector coupling, the electron, and the nucleon, respectively. This interaction can be generated by massive spin-1 boson exchange29, for example, from spontaneous breaking with two or more Higgs doublets. Some experiments on the interaction of neutrons have been conducted using polarized neutrons in liquid 4He30, combining experimental constraints from other experiments31 and polarized 3He coupling to the Earth32. While the parity-even spin- and velocity-dependent interaction for electrons has been experimentally investigated20,33,34,35,36, to the best of our knowledge, the parity-odd spin- and velocity-dependent interaction of electrons has been explored by few laboratory experiments such as electron-spin polarized torsion-pendulum experiments33,34 using the Sun and the Moon as a unpolarized test mass.

In the following, we explore the parity-odd spin- and velocity-dependent interaction of electrons using a SERF atomic magnetometer based on a similar experimental approach in refs. 20,21. The experiment sets a limit on the electron-nucleon coupling $$g_{\mathrm{A}}^{\mathrm{e}}g_{\mathrm{V}}^{\mathrm{N}} \, < \, 10^{ - 30}$$ for the boson mass less than 10−4 eV, equivalent to the interaction range larger than 10−3 m. This improves the current limit by up to 17 orders of magnitude.

## Results

### Exotic parity-odd spin- and velocity-dependent interaction

We investigated the interaction between polarized electrons and unpolarized nucleons based on a SERF atomic magnetometer:

$$V(\widehat {\mathbf{\sigma }}_i,{\mathbf{r}}) = g_{\mathrm{A}}^{\mathrm{e}}g_{\mathrm{V}}^{\mathrm{N}}\frac{\hbar }{{8\pi }}(2\widehat {\mathbf{\sigma }}_i \cdot {\mathrm{v}})\frac{{e^{ - r/\lambda }}}{r},$$
(1)

where is Planck’s constant, $$\widehat {\mathbf{\sigma }}_i$$ is the ith spin unit vector of the polarized electron, r is the separation vector in the direction between the polarized electron and the unpolarized nucleon, r = |r|, v is their relative velocity vector, and λ = (mbc)−1 is the interaction range (or the boson Compton wavelength) with mb being the boson mass and c being the speed of light in vacuum. $$g_{\mathrm{A}}^{\mathrm{e}}g_{\mathrm{V}}^{\mathrm{N}}$$ is the interaction coupling assuming no difference between electron-neutron and electron-proton couplings, and ignoring electron-electron couplings. The interaction induces the energy shift ΔE of the polarized electrons of the alkali-metal atoms in a SERF atomic vapor, which can be recast as

$$V = \Delta E = \gamma \hbar {\hat{\mathbf{\sigma }}}_i \cdot {\mathbf{B}}_{{\mathrm{eff}}},$$
(2)

where γ is the gyromagnetic ratio of the alkali-metal atoms and Beff is an effective magnetic field at the location of the atomic vapor, produced by the interaction V. From Eqs. (1) and (2), the effective magnetic field is

$${\mathbf{B}}_{{\mathrm{eff}}} = g_{\mathrm{A}}^{\mathrm{e}}g_{\mathrm{V}}^{\mathrm{N}}\frac{{2{\mathbf{v}}}}{{8\pi \gamma }}\frac{{e^{ - r/\lambda }}}{r}.$$
(3)

This experiment is to detect the Beff with a SERF atomic magnetometer.

The SERF atomic magnetometer contains two laser beams (for detail, see for example the description of a SERF magnetometer in ref. 37). The circularly polarized pump laser beam serves to pump atomic spins into the stretched state, creating nearly 100% polarization. The linearly polarized probe laser beam serves to detect the spin projection along the probe beam. The non-zero spin projection results in the rotation of light polarization, due to the strong birefringence of the spin-polarized vapor, and is measured with a polarizing beam splitter and two photo detectors as a small difference in the balanced output. The interaction of an external magnetic field with the polarized atomic spins leads to a change in the orientations of the spins and hence in the observed light polarization rotation. Because Eq. (2) formally describes the interaction of electron spins with the effective magnetic field, the interaction V causes similar changes in the spin orientation as the ordinary field, and can be probed with the magnetometer20,21.

### Experimental details

We employed a commercial cm-scale SERF magnetometer which is low cost, compact, and easy to operate38,39. It is based on a rubidium (Rb) 3 × 3 × 3 mm3 atomic vapor cell with a buffer gas. In order to operate the magnetometer in the SERF regime, the Rb vapor was heated to ~160 °C, which supplies sufficiently large Rb density (~1013 Rb atoms) as the source of optically polarized electrons. Furthermore, the magnetometer was placed inside a ferrite cylindrical shield (OD = 18 cm, height H = 38 cm, material thickness t = 0.6 cm) with end-caps and a three-layer open μ-metal co-axial cylinder (ID = 23 cm, OD = 29 cm, outer H = 69 cm) as shown in Fig. 1a. The magnetic shields sufficiently suppressed the external dc magnetic field and magnetic noise, but the residual fields inside the ferrite shield were compensated with three orthogonal coils [not shown in Fig. 1a]. The magnetometer was rigidly mounted inside the ferrite shield by a plastic holder.

For the source of unpolarized nucleons, we used a 2 × 2 × 2 cm3 non-magnetic bismuth germanate insulator (BGO), which contains a high nucleon density of 4.3 × 1024 cm−3. As shown in Fig. 1a, the BGO mass was connected to a stepper motor (LR43000 manufactured by Haydon Kerk Pittman), fixed on a three-axis translation stage (Thorlabs PT3), through a 1.5 m-long rigid fiberglass G10 cylindrical rod. The motor was positioned away from the magnetic shields and was enclosed by a single-layer open μ-metal box to suppress magnetic noise arising from the motor operation [see Fig. 1b]. The translation stage precisely adjusted the position of the mass in the vicinity of the Rb vapor. In order to prevent the mass from oscillating due to the movement of the long G10 rod, a non-magnetic air bushing (OAV0500IB) was closely mounted to the ferrite shield, providing an additional support of the G10 rod with nearly frictionless motion of the mass. To reduce systematic effects due to vibrations induced by the motor moving the mass, the frame holding both the atomic magnetometer and the magnetic shields was decoupled from the frame holding the moving mass, as indicated in Fig. 1a, b.

As illustrated in Fig. 1c, the relative velocity term in the interaction V was created by linearly moving the BGO mass along the z-axis with a constant velocity v next to the Rb vapor located in the magnetometer head. The standoff distance between the nearest surfaces of the mass and the vapor was set to 5 mm, limited by the location of the vapor. The Rb electron spins were oriented along the x-axis and the magnetometer was sensitive to the z component of a magnetic field with the intrinsic field noise level of 15 fT Hz−1/2, at low frequency. According to Eq. (3), the Beff is only along the z-axis in this configuration of the mass and the magnetometer because Beff is proportional to v.

Main sources of the systematic effects in this experiment are magnetic impurities buried inside the BGO mass generating fields measured on the order of 10−11 T and drift in the magnetometer signal on the order of 10−10 T. To suppress the systematic effects, the mass movement was alternated to subtract the magnetometer signals between upward v and downward −v mass motions, where v = |v|. This technique was highly effective since the sign of Beff is reversed for the opposite linear motions due to the single velocity term in Eq. (3), while the systematic effects are essentially the same20.

In Fig. 2, we present the interaction potential, written as

$$P_{{\mathrm{eff}}} = \frac{\hbar }{{8\pi }}(2\widehat {\mathbf{\sigma }}_i \cdot {\mathbf{v}})\frac{{e^{ - r/\lambda }}}{r},$$
(4)

as a function of the distance between the center of the BGO mass and the center of the Rb vapor cell, defined as Shift here, at different interaction ranges. The interaction potential was numerically calculated by a Monte Carlo integration20,21 which averaged the interaction potential Peff over the both volumes of the mass and the vapor. This was done by randomly generating 220 electron-nucleon pairs inside the volumes, summing the potential Peff between each pair based on Eq. (4) by assuming the interaction range, and then normalizing the resulting potential for the nucleon density of the mass. More details are described below. The velocity magnitude, v, was 15.38 mm s−1, used in the present experiments. As shown in Fig. 2, the interaction potential gets weaker as the mass moves away from the vapor and drops more drastically for the smaller interaction range due to the term of er/λ in Eq. (4). Since the interaction potential varies with the position of the mass with respect to the vapor, we selected one proper mass position near Shift = 0 in order to constrain the $$g_{\mathrm{A}}^{\mathrm{e}}g_{\mathrm{V}}^{\mathrm{N}}$$.

### Analysis

In the signal subtraction method to suppress the systematic effects, the BGO mass was extended along the z direction with the constant velocity of v = 15.38 mm s−1 for 0.325 s from the initial Shift of −1 mm to the final Shift of 4 mm, and then it was retracted toward the initial configuration with v = −15.38 mm s−1 for 0.325 s, as illustrated in Fig. 3a. The initial configuration was chosen because the sample holder contacted the head of the magnetometer when Shift < −1 mm. The motor was able to repeat this mass motion continuously, during which we recorded the magnetometer signals for 77 h using a 24-bit analog-to-digital converter (PXIe-4497 provided by National Instruments) with a sampling rate of 1 kHz.

Figure 3b shows standard time traces of the magnetometer signal, which present two full cycles of the mass motion reversal. Here one cycle refers to one extension with v = 15.38 mm s−1 and one retraction with v = −15.38 mm s−1 of the mass. The motor provided a voltage output which allows us to distinguish the mass motion’s direction, as shown in Fig. 3c. This was also recorded simultaneously with the magnetometer signal. The rising and falling edges served as the reference points for each half cycle. The motor had a delay time of 0.05 s after each extension and retraction of the mass, necessary to reverse the mass motion’s direction, thus the time elapsed for one cycle was 0.75 s. As discussed above, to extract the magnitude of the effective magnetic field, Beff, from the magnetometer signals, we selected the middle configuration of each motion which corresponds to Shift = 1.5 mm, as shown in Fig. 3a, equivalent to the data point at the first 50% of the data in each half cycle, as shown in Fig. 3b. However, instead of taking only one data point in each half cycle for the extraction of Beff, we used the mean of three data points, i.e., the data point at the first 50% of the data ± the adjacent data points. The distance between data points is 15 μm.

The mean value in each half cycle can be modeled as

$$D^ \pm (t) = \pm B_{{\mathrm{eff}}} + a_0 + a_1t + a_2t^2 + a_3t^3,$$
(5)

where the sign of + and − corresponds to the sign of the velocity, t is time, a0 is the dc offset field on the order of a few hundred pT, as shown in Fig. 3b, and the last three terms characterize the slow drifts in the magnetometer signal expanded in a polynomial function of time up to third order. These mainly arise from magnetometer electronics and temperature variations around the magnetometer. Note that a0 remains constant as the mass linear motion is reversed, unlike Beff. The magnetometer drift is mostly of first order: however, higher-order drifts can exist in the magnetometer signal due to, for example, opening/closing the laboratory door and other doors nearby the laboratory, or moving magnetic objects around the experimental setup. The simple difference between the first and the second half cycle with the time period of the half cycle, T, results in

$$D^ + (T) - D^ - (2T) = 2B_{{\mathrm{eff}}} - a_1T - 3a_2T^2 - 7a_3T^3.$$
(6)

Since this computation removes only the a0 term, the remaining drift terms can add systematic effects to the extraction of Beff. To this end, we used a more effective computation on the data which applies a [+1 −3 +3 −1] weighting to the mean value from each half cycle within two cycles20,40:

$$D^ + (T) - 3D^ - (2T) + 3D^ + (3T) - D^ - (4T) = 8B_{{\mathrm{eff}}} - 6a_3T^3.$$
(7)

This weighting computation effectively eliminates the dominant slow drift terms up to second order. The remaining third- or higher-order drifts might become major systematic effects if the data is averaged for a long time and the statistical uncertainty is reduced. According to Eq. (7), the Beff can be obtained by dividing the resulting value by a factor of 8 assuming that the remaining third order drift is negligible.

A histogram of Beff, extracted with the weighting computation from the magnetometer signals collected for 77 h, is shown in Fig. 4. A fit with a Gaussian distribution to the histogram gives Beff = (0.94 ± 2.15) × 10−16 T. According to Eq. (2), this equates to ΔE = (1.71 ± 3.90) × 10−20 eV with γ = 2π × 7.0 × 109 Hz T−137. The signal subtraction method together with the weighting computation suppresses the major systematic effects below the statistical sensitivity of 2.15 × 10−16 T.

To experimentally constrain the interaction V from our SERF magnetometer-based experiment, we performed a Monte Carlo integration for the interaction potential Peff, as described above. Based on Eqs. (1) and (2), the experimental limit to $$g_{\mathrm{A}}^{\mathrm{e}}g_{\mathrm{V}}^{\mathrm{N}}$$ was estimated from dividing the experimental sensitivity of ΔE of 3.90 × 10−20 eV by the interaction potential calculated at different interaction ranges20. Figure 5 shows the experimental limit to $$g_{\mathrm{A}}^{\mathrm{e}}g_{\mathrm{V}}^{\mathrm{N}}$$ in interaction ranges above 10−4 m (bottom axis), corresponding to the boson mass below 10−3 eV (top axis) with the data collection time of 77 h.

## Discussion

Our experiment sets an experimental limit on the parity-odd spin- and velocity-dependent interaction V for electrons in the boson mass ranges below 10−3 eV, which is essential for developing directions in WISPs searches. One great advantage in our experiment is that the boson mass range can be simultaneously scanned, unlike the resonant cavity-based WISP-searching experiments, such as ADMX, which require tuning experimental parameters for each axion mass under investigation. Using this experimental method, recently, we also set an experimental limit on the parity-even spin- and velocity-dependent interaction V4+5 between polarized electrons and unpolarized nucleons in the sub-meV range of the boson mass20. These experimental results demonstrate the feasibility of our experimental approach using the SERF atomic magnetometer. Our experiments will play an important role in exploring the uninvestigated spin-dependent interactions for polarized electrons21.

Further enhancement in the experimental sensitivity could be achievable. The SERF atomic magnetometer used in this work operates in a single-beam configuration in which a single circularly polarized laser beam is used to pump and probe the Rb electron spins39. The single-beam configuration is suboptimal in terms of magnetic field sensitivity. For an improved design, the magnetometer would use an orthogonal-beam configuration in which the probe beam is perpendicular to the pump beam, and the laser beams’ wavelengths are individually optimized. This modification can improve the sensitivity to 1 fT Hz−1/2 immediately. Further work will bring the magnetometer close to the fundamental photon shot noise limit of subfemto-Tesla to atto-Tesla41 by reducing the noise sources from laser frequency and intensity fluctuations and by using the hybrid optical pumping41.

In conclusion, we explored the exotic parity-odd spin- and velocity-dependent interaction V for polarized electrons based on an experimental approach utilizing a SERF atomic magnetometer. Our experiment sets an experimental limit on the interaction in the interaction range higher than 10−4 m corresponding to the boson mass lower than 10−3 eV.

## Methods

### Monte Carlo integration

The algorithm of the Monte Carlo integral is following:

1. (1)

220 random pairs of points inside both the volumes of the BGO mass and the Rb vapor cell are generated;

2. (2)

An interaction range λ between 1014 to 10−16 m is assumed to calculate the interaction potential $$P_{{\mathrm{eff}}}^i$$ between a randomly generated pair of points i,

$$P_{{\mathrm{eff}}}^i = \frac{\hbar }{{8\pi }}(2\widehat {\mathbf{\sigma }}_i \cdot {\mathbf{v}})\frac{{e^{ - r_i/\lambda }}}{{r_i}};$$
(8)
3. (3)

All the contributions to the potential are summed and normalized to give the average interaction potential for the nucleon density of BGO mass NBGO:

$$P_{{\mathrm{eff}}} = N_{{\mathrm{BGO}}}\frac{1}{{2^{20}}}\mathop {\sum}\limits_i^{2^{20}} {P_{{\mathrm{eff}}}^i} ;$$
(9)
4. (4)

The experimental limit to the coupling strength $$g_{\mathrm{A}}^{\mathrm{e}}g_{\mathrm{V}}^{\mathrm{N}}$$ is derived by dividing the experimental sensitivity of the energy shift ΔE for Rb atoms by the calculated average interaction potential;

5. (5)

The steps from (1) to (4) are repeated with another given interaction range between 10−1 to 10−6 m.

The Monte Carlo error can be ignored because the error numerically decreases as $$1/\sqrt N$$ where N is the number of random points, thus it is less than 1% with the 220 random points.

### SERF atomic magnetometer

A SERF atomic magnetometer employed for the experiments, manufactured by QuSpin Inc., is a compact, self-contained unit with all the necessary optical components and can be readily operated using a control software. Practically, the SERF regime is achieved by operating in a near-zero magnetic field and by heating the Rb vapor; a SERF magnetometer is sensitive at low frequency below 100 Hz. It is mainly composed of a 3 mm cube 87Rb vapor cell, a single semiconductor 795 nm laser for both optical pumping and probing, and silicon photodiodes39. The dimensions of the magnetometer module are: length 8 cm, width 1.4 cm, and height 2.1 cm. The cell temperature is around 160 °C for sufficiently large Rb density. The action of the laser beam creates a large number of polarized Rb electron spins in the vapor cell and its intensity transmitted through the cell is detected using the photodiode. The output from the photodoide is an analogue voltage signal and it can be converted to a magnetic field signal by applying a known calibration field to the vapor cell. In order to reduce the effects of the laser noise due to intensity fluctuation, the magnetometer uses a modulating field at 926 Hz39. The measured field at a certain low frequency is restored using a built-in lock-in amplifier and a low-pass filter.